Existence results for nonlinear degenerate elliptic equations with lower order terms

3.1 Proof of Theorem 2.1

The proof relies on an approximation procedure which is divided into several steps.

1. We prove the following a priori estimates:

   ∫Ω|un|t≤c0and∫Ω|∇un|q≤c0, (3.3)

   ∫Ω|∇Gk(un)|q≤ϵ(k), (3.4)

   for 1 ≤ q < q₁, where ϵ(k) does not depend on n and tends to zero when k tends to +∞.

   To prove the above estimates, we take φ = [1 − (1 + ∣u_n∣)^1−λ]sgn(u_n) as a test function in (3.2), where λ > 1 will be chosen later. Using Young’s inequality, it leads to

   α(λ−1)∫Ω|∇un|p(1+|un|)θ(p−1)+λ+ν∫Ω|un|t[1−(1+|un|)1−λ]≤α(λ−1)2∫Ω|∇un|p(1+|un|)θ(p−1)+λ+y11−rα2(λ−1)r/(1−r)∫Ω(1+|un|)θr(p−1)+λr1−r+∫Ω|f|+α(λ−1)4∫Ω|∇un|p(1+|un|)θ(p−1)+λ+(λ−1)4α1p−1∫Ω|F|pp−1. (3.5)

   Choosing S > 0 such that 1 − (1 + S)^1−λ = 1/2, we easily obtain

   12∫Ω|un|t≤12∫{x∈Ω:|un|≤S}|un|t+12∫{x∈Ω:|un|>S}|un|t≤12St|Ω|+∫Ω|un|t[1−(1+|un|)1−λ].

   The above inequality together with (3.5) imply that

   α(λ−1)4∫Ω|∇un|p(1+|un|)θ(p−1)+λ+ν2∫Ω|un|t≤y11−rα2(λ−1)r/(1−r)∫Ω(1+|un|)θr(p−1)+λr1−r+∫Ω|f|+ν2St|Ω|+(λ−1)4α1p−1∫Ω|F|pp−1. (3.6)

   Observing that t > [r + θ r(p − 1)]/(1 − r) (see (2.5)), one may choose λ in (3.6) such that [θ r(p − 1) + λ r]/(1 − r) < t, i.e. 1 < λ < [t(1 − r) − θ r(p − 1)]/r. Hence, we get

   ∫Ω|∇un|p(1+|un|)θ(p−1)+λ≤c1and∫Ω|un|t≤c1. (3.7)

   Therefore by the two estimates of (3.7), we infer that for q≤ptθ(p−1)+λ+t<ptθ(p−1)+1+t, it results (θ(p − 1) + λ)q/(p − q) ≤ t and so

   ∫Ω|∇un|q≤∫Ω|∇un|q(1+|un|)(θ(p−1)+λ)q/p(1+|un|)(θ(p−1)+λ)q/p≤c1qp∫Ω(1+|un|)(θ(p−1)+λ)q/(p−q)1−qp≤c2. (3.8)

   Setting q~=N(p−θ(p−1)−λ)N−θ(p−1)−λ and q0=N(p−θp+θ−1)N−θp+θ−1, obviously we have q̃ < q₀.

   If q₀ > 1, then by (2.1), (3.7) and taking 1 < λ < p − θ(p − 1), we find that for 1 ≤ q̃ < q₀,

   Cq~∫Ω|un|q~∗q~q~∗≤∫Ω|∇un|q~=∫Ω|∇un|q~(1+|un|)(θ(p−1)+λ)q~/p(1+|un|)(θ(p−1)+λ)q~p≤c1q~p∫Ω(1+|un|)q~∗1−q~p,

   where C is the Sobolev constant and q̃^∗ = (θ(p − 1) + λ) q̃/(p − q̃). By the choice of 1 < λ < p − θ(p − 1), we observe that q̃/q̃^∗ > 1 − q̃/p. Thus by Young’s inequality we easily get

   ∫Ω|∇un|q~≤c2for1≤q~<q0. (3.9)

   Taking 1 < λ < min{p − θ(p − 1), [t(1 − r) − θ r(p − 1)]/r}, by (3.7), (3.8) and (3.9), we conclude that (3.3) holds for 1 ≤ q < q₁ = max{ptt+1+θ(p−1),q0}.

   If q₀ ≤ 1, then (3.8) leads to (3.3) for 1 ≤ q < ptt+1+θ(p−1). Therefore, for both case q₀ ≤ 1 and q₀ > 1, we conclude that (3.3) holds true.

   Finally, the proof of (3.4) is similar at all to that of (3.3), except that the test function is changed to φ = [1 − (1 + ∣G_k(u_n)^1−λ∣)] sgn(G_k(u_n)), so we omit the detail here.

2. we now prove the following convergence result:

   ∥g(x,un)−g(x,u)∥L1(Ω)→0. (3.10)

   To do this, firstly it is easy to derive from the estimates (3.3) that for a subsequence of {u_n} (still denote by {u_n}), and a function u ∈ W01,q (Ω) ∩ L^t(Ω), such that as n → +∞ it results:

   un→u a.e. in Ωand weaklyinW01,q(Ω)∩Lt(Ω). (3.11)

   Then for fixed l, d > 0, taking T11d(|un|−l)+ sgn(u_n) as a test function in (3.2), we get by applying Young’s inequality and using (3.3),

   αd∫{x∈Ω:l≤|un|≤l+d}|∇un|p(1+|un|)θ(p−1)+∫{x∈Ω:|un|≥l+d}|g(x,un)|≤α2d∫{x∈Ω:l≤|un|≤l+d}|∇un|p(1+|un|)θ(p−1)+c3∫{x∈Ω:l≤|un|≤l+d}(1+|un|)θr(p−1)1−r+y∫{x∈Ω:|un|≥l+d}|∇un|pr+∫{x∈Ω:|un|≥l}|f|+α2d∫{x∈Ω:l≤|un|≤l+d}|∇un|p(1+|un|)θ(p−1)+c4∫{x∈Ω:l≤|un|≤l+d}|F|pt(p−1)(t−θ)t−θt,

   where c3=(1−r)(2drα)r/(1−r) and c4=1d(2pα)1p−1p−1pc0θt.

   By (2.5) and (3.3), we have

   c3∫A(l)(1+|un|)θr(p−1)1−r≤c3∫A(l)(1+|un|)tθr(p−1)t(1−r)|A(l)|t(1−r)−θr(p−1)t(1−r)≤c5|A(l)|t(1−r)−θr(p−1)t(1−r),

   where c5=2θr(p−1)1−rc0θr(p−1)t(1−r)c3 and A(l) = {x ∈ Ω : l ≤ ∣u_n∣ ≤ l + d}.

   The above two inequalities and (2.5) yield

   ∫Ω|g(x,un)|=∫Ω|g(x,un)|χ{x∈Ω:0≤|un(x)|≤l+d}+∫{x∈Ω:|un|≥l+d}|g(x,un)|≤∫Ω|g(x,un)|χ{x∈Ω:0≤|un(x)|≤l+d}+ε(l), (3.12)

   where the term ε(l) satisfying liml→+∞ ε(l) = 0 is denoted by

   ε(l)=y∥un∥W01,q(Ω)pr|{x∈Ω:|un|≥l+d}|q−prq+∫{x∈Ω:|un|≥l}|f|+c5|A(l)|t(1−r)−θr(p−1)t(1−r)+c4∫A(l)|F|pt(p−1)(t−θ)t−θt.

   By applying (3.12), Lebesgue’s dominated convergence theorem and Fatou lemma, it can be concluded that

   ∫Ω|g(x,un)|→∫Ω|g(x,u)|,asn→+∞. (3.13)

   Combining (3.13) with (3.11), we obtain the desired result (3.10).

3. End the proof.

   To do this, for any h > 0 let us take T_h(u_n) as a test function in (3.2), we easily obtain that

   ∫Ω|∇Th(un)|p≤c6hθ(p−1)+1+c7. (3.14)

   Then there exists a subsequence (still denoted {u_n}) such that

   Th(un)⇀Th(u)weaklyinW01,p(Ω). (3.15)

   Then use the argument of [20](see also [6, 8, 9]) one may get

   ∇un→∇ua.e.inΩ. (3.16)

   By (3.14)-(3.16) and Fatou lemma, we conclude

   ∫Ω|∇Th(u)|p≤c8and∫Ω|∇Th(un)−∇Th(u)|p≤c9, (3.17)

   which gives

   kp|Ω∖Ek,n|≤∫Ω∖Ek,n|∇Th(un)−∇Th(u)|p≤∫Ω|∇Th(un)−∇Th(u)|p≤c9, (3.18)

   where E_k,n := {x ∈ Ω : ∣∇ T_h(u_n)(x) − ∇ T_h(u)(x)∣ ≤ k}. Then we obtain

   ∫Ω|∇Th(un)−∇Th(u)|q=∫Ek,n|∇Th(un)−∇Th(u)|q+∫Ω∖Ek,n|∇Th(un)−∇Th(u)|q≤∫ΩχEk,n|∇Th(un)−∇Th(u)|q+c10kq−p,

   where we have used the results (3.16)-(3.18) and Hölder inequality. Thus, we get

   limn→+∞sup∫Ω|∇Th(un)−∇Th(u)|q≤c10kq−p∀k>0,

   which implies that

   ∫Ω|∇Th(un)−∇Th(u)|q→0n→+∞.

   Combining this result with (3.4), we deduce that

   un→ustronglyinW01,q(Ω). (3.19)

   Let n → ∞ in (3.2), by (2.2), (2.4), (2.5), (3.11) and (3.19), it follows that u is a weak solution of problem (𝒫) in the sense of Definition 2.1. Thus, the proof of Theorem 2.1 is finished.□

3.2 Proof of Theorem 2.2

1. we first prove the result when F = 0.

   Let us take φ = [(1 + ∣u_n∣)^δ − 1]sgn(u_n) in (3.2), where δ > 0 will be chosen later. We get after using Hölder’s inequality and Young’s inequality,

   αδ2∫Ω|∇un|p(1+|un|)δ−1−θ(p−1)+∫Ωg(x,un)[(1+|un|)δ−1]sgn(un)≤c~0∫Ω(1+|un|)δ−r(δ−1)+rθ(p−1)1−r+∥f∥Lm(Ω)∫Ω(1+|un|)δm′1m′. (3.20)

   Observing that as lims→+∞st[(1+s)δ−1](1+s)δ+t=1, there exists S₀ = S₀(δ, t) > 0 such that

   st[(1+s)δ−1]≥(1+s)δ+t2,∀s≥S0.

   Thus, by (2.5) and (3.20), we obtain

   ∫Ωg(x,un)[(1+|un|)δ−1]sgn(un)≥ν2∫{x∈Ω:|un|>S0}(1+|un|)δ+t. (3.21)

   By (3.20), (3.21) and notice that t > [r + θ r(p − 1)]/(1 − r), we infer that

   αδ2∫Ω|∇un|p(1+|un|)δ−1−θ(p−1)+ν2∫{x∈Ω:|un|>S0}(1+|un|)δ+t≤c~0∫{x∈Ω:|un|≤S0}(1+|un|)δ−r(δ−1)+rθ(p−1)1−r+∫{x∈Ω:|un|>S0}(1+|un|)δ−r(δ−1)+rθ(p−1)1−r+∥f∥Lm(Ω)∫Ω(1+|un|)δm′1m′≤v4∫{x∈Ω:|un|>S0}(1+|un|)δ+t+c~1+∥f∥Lm(Ω)∫Ω(1+|un|)δm′1m′. (3.22)

   By choosing δ m^′ = δ + t (i.e.δ = t(m − 1) in (3.22), we have

   αδ2∫Ω|∇un|p(1+|un|)δ−1−θ(p−1)+ν4∫{x∈Ω:|un|>S0}(1+|un|)tm≤ν8∫{x∈Ω:|un|>S0}(1+|un|)tm+c~2, (3.23)

   which implies that

   ∫Ω|∇un|p(1+|un|)t(m−1)−1−θ(p−1)≤c~3and∫Ω(1+|un|)tm≤c~3.

   Obversely, we have

   ∫Ω|∇un|p≤c~3,if t(m−1)−θ(p−1)−1≥0. (3.24)

   Similarly to (3.8), in the case t(m − 1) − θ(p − 1) − 1 < 0 we get

   ∫Ω|∇un|q4≤c~4,withq4=min{p,ptmt+1+θ(p−1)}. (3.25)

   To end this proof in case F = 0, we shall choose a different δ in (3.22). Indeed, similar to (3.21) we infer that for S₁(independent of n) large enough,

   αδ2∫Ω|∇un|p(1+|un|)δ−1−θ(p−1)=ppαδ2[δ−1−θ(p−1)+p]p∫Ω∇(1+|un|)δ−1−θ(p−1)+pp−1p≥c~5∫|un|>S1(1+|un|)[δ−1−θ(p−1)+p]p∗ppp∗. (3.26)

   By (3.22) and (3.26), it follows that

   ∫Ω(1+|un|)[δ−1−θ(p−1)+p]p∗ppp∗≤c~6+c~6∫Ω(1+|un|)δm′1m′. (3.27)

   Let us choose δm′=[δ−1−θ(p−1)+p]p∗p(i.e.δ=p∗(p−1)(1−θ)m′p−p∗), then we have δm′=m′p∗(p−1)(1−θ)m′p−p∗. Hence, applying Young’s inequality in (3.27), we get

   ∫Ω|un|m′p∗(p−1)(1−θ)m′p−p∗≤c~7. (3.28)

   Substituting (3.28) into (3.22), we find

   ∫Ω|∇un|p(1+|un|)δ−1−θ(p−1)≤c~8,δ=p∗(p−1)(1−θ)m′p−p∗. (3.29)

   In the case that δ ≥ 1 + θ(p − 1) (i.e. m ≥ Np(1−θ)[N(p−1)+p]+p2θ), we have

   ∫Ω|∇un|p≤c~8, (3.30)

   while in the case that δ < 1 + θ(p − 1)(that is m < Np(1−θ)[N(p−1)+p]+p2θ), combining (3.28) with (3.29) and arguing as in (3.8) we obtain

   ∫Ω|∇un|q5≤c~9withq5=pm′p∗(p−1)(1−θ)(p−1)[p∗(m′−1)+θm′(p−p∗)]+m′p−p∗. (3.31)

   Finally, by (3.24), (3.25), (3.28), (3.30) and (3.31), arguing as in the step 2 and step 3 we conclude that the result of this theorem holds true when F = 0.

   If F ≠ 0, then in the right-hand side of (3.20) it contains also the following term

   δ∫ΩF(1+|un|)δ−1∇un.

   Using Young’s inequality, we easily obtain that

   δ∫ΩF(1+|un|)δ−1∇un≤αδ4∫Ω|∇un|p(1+|un|)δ−1−θ(p−1)+δ4α1p−1∫Ω|F|pp−1(1+|un|)δ+θ−1.

   Thus if δ ≤ 1 − θ we obtain

   δ4α1p−1∫Ω|F|pp−1(1+|un|)δ+θ−1≤δ4α1p−1∫Ω|F|pp−1,

   while if δ > 1 − θ we obtain

   δ4α1p−1∫Ω|F|pp−1(1+|un|)δ+θ−1≤ν16∫Ω(1+|un|)δ+t+c~10∫Ω|F|pp−1δ+tδ+θ−1′.

   As we have said in Remark 2.1, to guarantee the convergence result (3.10), we should assume F ∈ (Lpt(p−1)(t−θ)(Ω))N. Observe that the assumption b ≥ b₀ is equivalent to require that if δ ≤ 1 − θ with δ = t(m − 1), then F ∈ (Lpp−1(Ω))N; while if δ > 1 − θ then F∈(Lptm(p−1)(t+1−θ)(Ω))N, where ptm(p−1)(t+1−θ)=pp−1δ+tδ+θ−1′. Similarly, the assumption b ≥ b₁ is equivalent to require that if δ ≤ 1 − θ with δ=p∗(p−1)(1−θ)m′p−p∗, then F∈(Lpp−1(Ω))N; while if δ > 1 − θ then F∈(Lpp−1δ+tδ+θ−1′(Ω))N. Thus if b ≥ b₀, using the previous estimates and proceeding as in the case F = 0, we conclude that u_n is equibounded in W01,q2 (Ω) ∩ L^tm(Ω); while if b > b₁ we can conclude that u_n is equibounded in W01,q3(Ω)∩Lm′p∗(p−1)(1−θ)m′p−p∗(Ω). Hence the assert follows as before. This completes the proof.□